Figure 22.23 shows some equipotentials in the x9y plane. (a) In what region is the

Why can a bird perch on a high-voltage power line without getting electrocuted?

One proton is accelerated from rest by a uniform electric field, another proton by a nonuniform electric field. If they move through the same potential difference, how do their final speeds compare?

Would a free electron move toward higher or lower potential?

The electric field at the center of a uniformly charged ring is obviously zero, yet Example 22.6 shows that the potential at the center isnt zero. How is this possible?

Must the potential be zero at any point where the electric field is zero? Explain.

Must the electric field be zero at any point where the potential is zero? Explain.

The potential is constant throughout an entire volume. What must be true of the electric field within that volume?

In considering the potential of an infinite flat sheet, why isnt it useful to take the zero of potential at infinity?

Cherry picker trucks for working on power lines often carry electrocution hazard signs. Explain how this hazard arises and why it might be more of a danger to someone on the ground than to a worker on the truck.

Can equipotential surfaces intersect? Explain.

Is the potential at the center of a hollow, uniformly charged spherical shell higher than, lower than, or the same as at the surface?

A solid sphere contains positive charge uniformly distributed throughout its volume. Is the potential at its center higher than, lower than, or the same as at the surface?

Two equal but opposite charges form a dipole. Describe the equipotential surface on which V = 0.

The electric potential in a region increases linearly with distance. What can you conclude about the electric field in this region?

How much work does it take to move a 50@C charge against a 12-V potential difference?

The potential difference between the two sides of an ordinary electric outlet is 120 V. How much energy does an electron gain when it moves from one side to the other?

It takes 45 J to move a 15-mC charge from point A to point B. Whats the potential difference VAB?

Show that 1 V/m is the same as 1 N/C.

Find the magnitude of the potential difference between two points located 1.4 m apart in a uniform 650-N/C electric field, if a line between the points is parallel to the field.

A charge of 3.1 C moves from the positive to the negative terminal of a 9.0-V battery. How much energy does the battery impart to the charge?

A proton, an alpha particle (a bare helium nucleus), and a singly ionized helium atom are accelerated through a 100-V potential difference. How much energy does each gain?

The potential difference across a typical cell membrane is about 80 mV. How much work is done on a singly ionized potassium ion moving through the membrane?

An electric field is given by ES= E0 nj, where E0 is a constant.Find the potential as a function of position, taking V = 0 aty = 0.

The classical picture of the hydrogen atom has the electron orbiting0.0529 nm from the proton. Whats the electric potentialassociated with the protons electric field at this distance?

The potential at the surface of a 10-cm-radius sphere is 4.8 kV.Whats the spheres total charge, assuming charge is distributedin a spherically symmetric way?

Youre developing a switch for high-voltage power lines. Thesmallest part in your design is a 5.0-cm-diameter metal sphere.What do you specify for the maximum potential on your switch if the electric field at the spheres surface isnt to exceed the3-MV/m breakdown field of air?

A 3.5-cm-diameter isolated metal sphere carries 0.86 C. (a)Find the potential at the spheres surface. (b) If a proton were releasedfrom rest at the surface, what would be its speed far from the sphere?

In a uniform electric field, equipotential planes that differ by 5.00 V are 2.54 cm apart. Whats the field strength?

Figure 22.22 shows a plot of potential versus position along the x-axis. Make a plot of the x-component of the electric field for this situation.

Figure 22.23 shows some equipotentials in the x9y plane. (a) In what region is the electric field strongest? What are (b) the direction and (c) the magnitude of the field in this region?

T h e e l e c t r i c p o t e n t i a l i n a r e g i o n i s g iv e n b y V = 2xy - 3zx + 5y2 , with V in volts and the coordinates in meters. Find (a) the potential and (b) the components of the electric field at the point x = 1 m, y = 1 m, z = 1 m

Dielectric breakdown of air occurs at fields of 3 MV/m. Find (a) the maximum potential (measured from infinity) for the sphere of Example 22.3 before dielectric breakdown occurs at the spheres surface and (b) the charge on the sphere at this potential.

Youre an automotive engineer working on the ignition system for a new engine. Its spark plugs have center electrodes made from 2.0-mm-diameter wire. The electrode ends gradually wear to a hemispherical shape, so they behave approximately like charged spheres. Your job is to specify the minimum potential that ensures these plugs will spark in air, neglecting the presence of the second electrode.

A large metal sphere has three times the diameter of a smaller sphere and carries three times the charge. Both spheres are isolated, so their surface charge densities are uniform. Compare (a) the potentials (relative to infinity) and (b) the electric field strengths at their surfaces.

Two points A and B lie 15 cm apart in a uniform electric field, with the path AB parallel to the field. If the potential difference VAB is 840 V, whats the field strength?

The electric field within a cell membrane is approximately 8.0 MV/m and is essentially uniform. If the membrane is 10 nm thick, whats the potential difference across the membrane?

Whats the potential difference between the terminals of a battery that can impart 7.2*10-19 J to each electron that moves between the terminals?

Whats the charge on an ion that gains 1.6*10-15 J when it moves through a potential difference of 2500 V?

Two flat metal plates are a distance d apart, where d is small compared with the plate size. If the plates carry surface charge densities{s, show that the magnitude of the potential difference between them is V = sd/e0.

An electron passes point A moving at 6.5 Mm/s. At point B it comes to a stop. Find the potential difference VAB.

A 5.0-g object carries 3.8 C. It acquires speed v when accelerated from rest through a potential difference V. If a 2.0-g object acquires twice the speed under the same circumstances, whats its charge?

Points A and B lie 32.0 cm apart on a line extending radially from a point charge Q, and the potentials at these points are VA = 362 V and VB = 146 V. Find Q and the distance r between point A and the charge.

A sphere of radius R carries negative charge of magnitude Q, distributed in a spherically symmetric way. Find an expression for the escape speed for a proton at the spheres surfacethat is, the speed that would enable the proton to escape to arbitrarily large distances starting at the spheres surface

Proton-beam therapy can be preferable to X rays for cancer treatment (although much more expensive) because protons deliver most of their energy to the tumor, with less damage to healthy tissue. A cyclotron used to accelerate protons for cancer treatment repeatedly passes the protons through a 15-kV potential difference. (a) How many passes are needed to bring the protons kinetic energy to 1.2*10-11 J? (b) Whats that energy in eV?

A thin spherical shell has radius R and total charge Q distributed uniformly over its surface. Find the potential at its center.

A solid sphere of radius R carries charge Q distributed uniformly throughout its volume. Find the potential difference from the spheres surface to its center. (Hint: Consult Example 21.1.)

Find the potential as a function of position in the electric field E S = axni, where a is a constant and where youre taking V = 0 at x = 0.

Your radio station needs a new coaxial cable to connect the transmitter and antenna. One possible cable consists of a 2.0-mmdiameter inner conductor and an outer conductor with diameter 1.6 cm and negligible thickness (Fig. 22.24); the maximum safe potential difference between the conductors is 2 kV. In your application, the conductors carry charge densities {62 nC/m. Will this cable work for you?

The potential difference between the surface of a 3.0- cm- diameter power line and a point 1.0 m distant is 3.9 kV. Find the line charge density on the power line.

Three equal charges q form an equilateral triangle of side a. Find the potential, relative to infinity, at the center of the triangle.

A charge +Q lies at the origin and -3Q at x = a. Find two points on the x-axis where V = 0.

Two identical charges q lie on the x-axis at{a. (a) Find an expression for the potential at all points in the xy plane. (b) Show that your result reduces to the potential of a point charge for distances large compared with a.

A dipole of moment p = 2.9 nC # m consists of two charges separated by far less than 10 cm. Find the potential 10 cm from the dipole (a) on its axis, (b) at 45 to its axis, and (c) on its perpendicular bisector.

A thin plastic rod 20 cm long carries 3.2 nC distributed uniformly over its length. (a) If the rod is bent into a ring, find the potential at its center. (b) If its bent into a semicircle, find the potential at the center of the semicircle.

A thin ring of radius R carries charge 3Q distributed uniformly over three-fourths of its circumference, and -Q over the rest. Find the potential at the rings center.

The potential at the center of a uniformly charged ring is 45 kV, and 15 cm along the ring axis the potential is 33 kV. Find the rings radius and total charge.

The annulus shown in Fig. 22.25 carries a uniform surface charge density s. Find an expression for the potential at an arbitrary point P on its axis.

The potential in a region is given by V = axy, where a is a constant.(a) Determine the electric field in the region. (b) Sketchsome equipotentials and field lines.

Use Equation 22.6 to calculate the electric field on the perpendicularbisector of a dipole, and show that your result is equivalentto Equation 20.6a.

Use the result of Example 22.6 to determine the on-axis field ofa charged ring, and verify that your answer agrees with the resultof Example 20.6.

The electric potential in a region is given by V = -V01r/R2,where V0 and R are constants and r is the radial distance from theorigin. Find expressions for the magnitude and direction of the electric field in this region.

Two metal spheres each 1.0 cm in radius are far apart. One sphere carries 38 nC, the other -10 nC. (a) Whats the potential on each? (b) If the spheres are connected by a thin wire, what will be the potential on each once equilibrium is reached? (c) How much charge moves between the spheres in order to achieve equilibrium?

Two 5.0-cm-diameter conducting spheres are 8.0 m apart, and each carries 0.12 C. Determine (a) the potential on each sphere, (b) the field strength at the surface of each sphere, (c) the potential midway between the spheres, and (d) the potential difference between the spheres.

A 2.0-cm-radius metal sphere carries 75 nC and is surrounded by a concentric spherical conducting shell of radius 10 cm carrying -75 nC. (a) Find the potential difference between shell and sphere. (b) How would your answer change if the shells charge were +150 nC?

A sphere of radius R carries a nonuniform but spherically symmetric volume charge density that results in an electric field in the sphere given by E S = E01r/R22nr, where E0 is a constant. Find the potential difference from the spheres surface to its center

The potential as a function of position in a region is given by V1x2 = 3x - 2x 2 - x 3 , with x in meters and V in volts. Find (a) all points on the x-axis where V = 0, (b) an expression for the electric field, and (c) all points on the x-axis where E = 0.

A conducting sphere 5.0 cm in radius carries 60 nC. Its surrounded by a concentric spherical conducting shell of radius 15 cm carrying -60 nC. (a) Find the potential at the spheres surface, taking V = 0 at infinity. (b) Repeat for the case when the shell carries +60 nC

Show that the result of Example 22.8 approaches the field of a point charge for x W a. (Hint: Youll need to apply the binomial approximation from Appendix A to the expression 1/2x 2 + a2 .)

The potential on the axis of a uniformly charged disk at 5.0 cm from the disk center is 150 V; the potential 10 cm from the disk center is 110 V. Find the disk radius and its total charge.

A uranium nucleus (mass 238 u, charge 92e) decays, emitting an alpha particle (mass 4 u, charge 2e) and leaving a thorium nucleus (mass 234 u, charge 90e). At the instant the alpha particle leaves the nucleus, the centers of the two are 7.4 fm apart and essentially at rest. Treating each particle as a spherical charge distribution, find their speeds when theyre a great distance apart.

The Taser, an ostensibly nonlethal weapon used by police to subdue unruly suspects, shoots two conducting darts into the victims body. Thin wires connect the darts back to the weapon, and once the darts are embedded the weapon applies a 1200-V potential across them and delivers short pulses of electric charge to the darts. Each pulse carries 100 C of charge. How much energy does the weapon need to supply to each charge pulse as it moves through the body from one dart to the other?

Using the dipole potential at points far from a dipole (given by Equation 22.6 in Example 22.5), show that the electric field at an arbitrary position can be written E S = kp r3 c13cos2u - 12ni + 3sin u cos uj n d where the x-axis coincides with the dipole axis.

Measurements of the potential at points on the axis of a charged disk are given in the two tables below, one for measurements made close to the disk and the other for measurements made far away. In both tables x is the coordinate measured along the disk axis with the origin at the disk center, and the zero of potential is taken at infinity. (a) For each set of data, determine a quantity that, when you plot potential against it, should yield a straight line. Make your plots, establish a best-fit line, and determine its slope. Use your slopes to find (b) the total charge on the disk and (c) the disk radius. (Hint: Consult Example 22.7.) Table 1 x (mm) 2.0 4.0 6.0 8.0 10.0 V (V) 900 876 843 820 797 Table 2 x (cm) 20 30 40 60 100 V (V) 165 118 80 58 30

Find an equation describing the V = 0 equipotential in the x-y plane for the situation of Conceptual Example 22.1. That is, find a relation between x and y that holds on this equipotential.

A thin rod of length L carries charge Q distributed uniformly over its length. (a) Show that the potential in the plane that perpendicularly bisects the rod is given by V(r) = 2kQ L ln c L 2r + A1 + L2 4r2 d where r is the perpendicular distance from the rod center and where the zero of potential is taken at infinity. (b) Show that this expression reduces to an expected result when r W L. (Hint: See Appendix A for a series expansion of the logarithm.)

For the rod of the preceding problem, (a) find an expression for the magnitude of the electric field in the perpendicular bisecting plane as a function of the distance r from the rod center. (b) Show that your expression reduces to an expected result when r W L. (c) Whats the direction of the field?

A disk of radius a carries nonuniform surface charge density s = s01r/a2, where s0 is a constant. (a) Find the potential at an arbitrary point x on the disk axis, where x = 0 is the disk center. (b) Use the result of (a) to find the electric field on the disk axis, and (c) show that the field reduces to an expected form for x W a.

An open-ended cylinder of radius a and length 2a carries charge q spread uniformly over its surface. Find the potential at the center of the cylinder. (Hint: Treat the cylinder as a stack of charged rings, and integrate.)

A line charge extends along the x-axis from -L/2 to L/2. Its line charge density is l = l01x/L22 , where l0 is a constant. Find an expression for the potential on the x-axis for x 7 L/2. Check that your expression reduces to an expected result for x W L.

Repeat Problem 79 for the charge distribution l = l0 x/L. (Hint: What does this charge distribution resemble at large distances?)

Youre sizing a new electric transmission line, and you can save money with thinner wire. The potential difference between the line and the ground, 60 m below, is 115 kV. The field at the wire surface cannot exceed 25% of the 3-MV/m breakdown field in air. Neglecting charges in the ground itself, what minimum wire diameter do you specify? (Hint: Youll have to do a numerical calculation.)

BIO Standard electrocardiography measures time-dependent potential differences between multiple points on the body, giving cardiologists multiple perspectives on the hearts electrical activity. In contrast, Fig. 22.26 is a snapshot showing a more detailed picture at an instant of time. The lines are equipotentials on the surface of a human torso, associated with the hearts electrical activity. Relative to the line marked V = 0, the potential is negative to the upper left (black) and positive to the lower right (color). From the equipotentials, you can infer that the hearts electrical structure resembles that of a a. uniform charged sheet. b. dipole. c. point charge. d. uniformly charged sphere

BIO Standard electrocardiography measures time-dependent potential differences between multiple points on the body, giving cardiologists multiple perspectives on the hearts electrical activity. In contrast, Fig. 22.26 is a snapshot showing a more detailed picture at an instant of time. The lines are equipotentials on the surface of a human torso, associated with the hearts electrical activity. Relative to the line marked V = 0, the potential is negative to the upper left (black) and positive to the lower right (color). The electric field in the vicinity of the heart points approximately a. from upper left to lower right. b. from lower left to upper right. c. from upper right to lower left. d. from lower right to upper left

BIO Standard electrocardiography measures time-dependent potential differences between multiple points on the body, giving cardiologists multiple perspectives on the hearts electrical activity. In contrast, Fig. 22.26 is a snapshot showing a more detailed picture at an instant of time. The lines are equipotentials on the surface of a human torso, associated with the hearts electrical activity. Relative to the line marked V = 0, the potential is negative to the upper left (black) and positive to the lower right (color). The electric field is strongest in the region marked a. A. b. B. c. C. d. D

BIO Standard electrocardiography measures time-dependent potential differences between multiple points on the body, giving cardiologists multiple perspectives on the hearts electrical activity. In contrast, Fig. 22.26 is a snapshot showing a more detailed picture at an instant of time. The lines are equipotentials on the surface of a human torso, associated with the hearts electrical activity. Relative to the line marked V = 0, the potential is negative to the upper left (black) and positive to the lower right (color). The electric field in region A is approximately a. 20 N/C. b. 2 mN/C. c. 20 mN/C. d. 2 kN/C.